scholarly journals Complexity issues of perfect secure domination in graphs

2021 ◽  
Vol 55 ◽  
pp. 11
Author(s):  
P. Chakradhar ◽  
P. Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Split Graphs ◽  
Secure Domination ◽  
Convex Bipartite Graphs

Let G = (V, E) be a simple, undirected and connected graph. A dominating set S is called a secure dominating set if for each u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E and (S \{v}) ∪{u} is a dominating set of G. If further the vertex v ∈ S is unique, then S is called a perfect secure dominating set (PSDS). The perfect secure domination number γps(G) is the minimum cardinality of a perfect secure dominating set of G. Given a graph G and a positive integer k, the perfect secure domination (PSDOM) problem is to check whether G has a PSDS of size at most k. In this paper, we prove that PSDOM problem is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs, planar graphs and dually chordal graphs. We propose a linear time algorithm to solve the PSDOM problem in caterpillar trees and also show that this problem is linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. Finally, we show that the domination and perfect secure domination problems are not equivalent in computational complexity aspects.

Total vertex-edge domination in graphs: Complexity and algorithms

2021 ◽  
Author(s):  
Nitisha Singhwal ◽  
Palagiri Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Linear Programming Formulation ◽  
Convex Bipartite Graphs

Let [Formula: see text] be a simple, undirected and connected graph. A vertex [Formula: see text] of a simple, undirected graph [Formula: see text]-dominates all edges incident to at least one vertex in its closed neighborhood [Formula: see text]. A set [Formula: see text] of vertices is a vertex-edge dominating set of [Formula: see text], if every edge of graph [Formula: see text] is [Formula: see text]-dominated by some vertex of [Formula: see text]. A vertex-edge dominating set [Formula: see text] of [Formula: see text] is called a total vertex-edge dominating set if the induced subgraph [Formula: see text] has no isolated vertices. The total vertex-edge domination number [Formula: see text] is the minimum cardinality of a total vertex-edge dominating set of [Formula: see text]. In this paper, we prove that the decision problem corresponding to [Formula: see text] is NP-complete for chordal graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. The problem of determining [Formula: see text] of a graph [Formula: see text] is called the minimum total vertex-edge domination problem (MTVEDP). We prove that MTVEDP is linear time solvable for chain graphs and threshold graphs. We also show that MTVEDP can be approximated within approximation ratio of [Formula: see text]. It is shown that the domination and total vertex-edge domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MTVEDP is presented.

scholarly journals Algorithmic Aspects of Some Variants of Domination in Graphs

2020 ◽  
Vol 28 (3) ◽  
pp. 153-170
Author(s):  
J. Pavan Kumar ◽  
P.Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Bipartite Graphs ◽  
Minimum Size ◽  
Induced Subgraph ◽  
Split Graphs ◽  
Np Complete ◽  
Secure Domination

AbstractA set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E, i.e., N[S] = V . A dominating set S is an isolate dominating set (IDS) if the induced subgraph G[S] has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set S ⊆ V is an independent set if G[S] has no edge. A set S ⊆ V is a secure dominating set of G if, for each vertex u ∈ V \ S, there exists a vertex v ∈ S such that (u, v) ∈ E and (S \ {v}) ∪ {u} is a dominating set of G. In addition, we initiate the study of a new domination parameter called, independent secure domination. A set S ⊆ V is an independent secure dominating set (InSDS) if S is an independent set and a secure dominating set of G. The minimum size of an InSDS in G is called the independent secure domination number of G and is denoted by γis(G). Given a graph G and a positive integer k, the InSDM problem is to check whether G has an independent secure dominating set of size at most k. We prove that InSDM is NP-complete for bipartite graphs and linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. The MInSDS problem is to find an independent secure dominating set of minimum size, in the input graph. Finally, we show that the MInSDS problem is APX-hard for graphs with maximum degree 5.

scholarly journals Semipaired Domination in Some Subclasses of Chordal Graphs

2021 ◽  
Vol vol. 23 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Michael A. Henning ◽  
Arti Pandey ◽  
Vikash Tripathi
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Positive Side ◽  
Domination Problem ◽  
Np Complete ◽  
Decision Version

A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.

Algorithmic aspects of k-part degree restricted domination in graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050057
Author(s):  
S. S. Kamath ◽  
A. Senthil Thilak ◽  
M. Rashmi
Keyword(s):  
Communication Networks ◽  
Polynomial Time ◽  
Dominating Set ◽  
Domination Number ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Circle Graphs ◽  
Minimum Cardinality

The concept of network is predominantly used in several applications of computer communication networks. It is also a fact that the dominating set acts as a virtual backbone in a communication network. These networks are vulnerable to breakdown due to various causes, including traffic congestion. In such an environment, it is necessary to regulate the traffic so that these vulnerabilities could be reasonably controlled. Motivated by this, [Formula: see text]-part degree restricted domination is defined as follows. For a positive integer [Formula: see text], a dominating set [Formula: see text] of a graph [Formula: see text] is said to be a [Formula: see text]-part degree restricted dominating set ([Formula: see text]-DRD set) if for all [Formula: see text], there exists a set [Formula: see text] such that [Formula: see text] and [Formula: see text]. The minimum cardinality of a [Formula: see text]-DRD set of a graph [Formula: see text] is called the [Formula: see text]-part degree restricted domination number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we present a polynomial time reduction that proves the NP -completeness of the [Formula: see text]-part degree restricted domination problem for bipartite graphs, chordal graphs, undirected path graphs, chordal bipartite graphs, circle graphs, planar graphs and split graphs. We propose a polynomial time algorithm to compute a minimum [Formula: see text]-DRD set of a tree and minimal [Formula: see text]-DRD set of a graph.

scholarly journals Graphs with equal domination and covering numbers

2019 ◽  
Vol 39 (1) ◽  
pp. 55-71 ◽  
Author(s):  
Andrzej Lingas ◽  
Mateusz Miotk ◽  
Jerzy Topp ◽  
Paweł Żyliński
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Minimum Cardinality ◽  
Covering Numbers ◽  
Intersection Points ◽  
Covering Set ◽  
Appl Math ◽  
Side Result

Abstract A dominating set of a graph G is a set $$D\subseteq V_G$$D⊆VG such that every vertex in $$V_G-D$$VG-D is adjacent to at least one vertex in D, and the domination number $$\gamma (G)$$γ(G) of G is the minimum cardinality of a dominating set of G. A set $$C\subseteq V_G$$C⊆VG is a covering set of G if every edge of G has at least one vertex in C. The covering number $$\beta (G)$$β(G) of G is the minimum cardinality of a covering set of G. The set of connected graphs G for which $$\gamma (G)=\beta (G)$$γ(G)=β(G) is denoted by $${\mathcal {C}}_{\gamma =\beta }$$Cγ=β, whereas $${\mathcal {B}}$$B denotes the set of all connected bipartite graphs in which the domination number is equal to the cardinality of the smaller partite set. In this paper, we provide alternative characterizations of graphs belonging to $${\mathcal {C}}_{\gamma =\beta }$$Cγ=β and $${\mathcal {B}}$$B. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to $${\mathcal {B}}$$B, and, as a side result, we conclude that the algorithm of Arumugam et al. (Discrete Appl Math 161:1859–1867, 2013) allows to recognize all the graphs belonging to the set $${\mathcal {C}}_{\gamma =\beta }$$Cγ=β in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that it can be solved in $$O(n \log n + m)$$O(nlogn+m) time, where n is the number of line segments of the input grid and m is the number of its intersection points.

scholarly journals A linear time algorithm for minimum equitable dominating set in trees

2021 ◽  
Vol 40 (4) ◽  
pp. 805-814
Author(s):  
Sohel Rana ◽  
Sk. Md. Abu Nayeem
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Minimum Cardinality ◽  
Np Complete ◽  
General Graphs

Let G = (V, E) be a graph. A subset De of V is said to be an equitable dominating set if for every v ∈ V \ De there exists u ∈ De such that uv ∈ E and |deg(u) − deg(v)| ≤ 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γe. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum equitable dominating set of a tree.

On computing secure domination of trees

2020 ◽  
pp. 2150055
Author(s):  
Abolfazl Poureidi
Keyword(s):  
Linear Time ◽  
Search Algorithm ◽  
Dominating Set ◽  
Domination Number ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Time And Space ◽  
Minimum Cardinality ◽  
Secure Domination ◽  
Appl Math

Let [Formula: see text] be a graph. A subset [Formula: see text] is a dominating set of [Formula: see text] if for each [Formula: see text] there is a vertex [Formula: see text] adjacent to [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is a secure dominating set of [Formula: see text] if for each [Formula: see text] there is a vertex [Formula: see text] adjacent to [Formula: see text] such that [Formula: see text] is also a dominating set of [Formula: see text]. The minimum cardinality of a secure dominating set of [Formula: see text] is called the secure domination number of [Formula: see text]. Burger et al. [A linear algorithm for secure domination in trees, Discrete Appl. Math. 171 (2014) 15–27] proposed a nontrivial algorithm for computing a minimum secure dominating set of a given tree in linear time and space. In this paper, we give a dynamic programming algorithm to compute the secure domination number of a given tree [Formula: see text] in [Formula: see text] time and space and then using a backtracking search algorithm we can find a minimum secure dominating set of [Formula: see text] in [Formula: see text] time and space that its implementation is much simpler than the implementation of the algorithm proposed by Burger et al.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

Fast and simple algorithms for counting dominating sets in distance-hereditary graphs

2020 ◽  
pp. 2150012
Author(s):  
Min-Sheng Lin
Keyword(s):  
Linear Time ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Dominating Sets ◽  
Complete Problem ◽  
Split Graphs ◽  
Linear Time Algorithms ◽  
Chordal Bipartite Graphs

Counting dominating sets (DSs) in a graph is a #P-complete problem even for chordal bipartite graphs and split graphs, which are both subclasses of weakly chordal graphs. This paper investigates this problem for distance-hereditary graphs, which is another known subclass of weakly chordal graphs. This work develops linear-time algorithms for counting DSs and their two variants, total DSs and connected DSs in distance-hereditary graphs.

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